Electronic Journal of Probability

Asymptotics of the rising moments for the coupon collector's problem

Aristides Doumas and Vassilis Papanicolaou

Full-text: Open access


We develop techniques of computing the asymptotics of the moments of the number $T_N$ of coupons that a collector has to buy in order to find all $N$ existing different coupons as $N\rightarrow \infty.$ The probabilities (occurring frequencies) of the coupons can be quite arbitrary. After mentioning the case where the coupon probabilities are equal we consider the general case (of unequal probabilities). For a large class of distributions (after adopting a dichotomy) we arrive at the leading behavior of the moments of $T_N$ as $N\rightarrow \infty.$ We also present various illustrative examples.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 41, 15 pp.

Accepted: 22 March 2013
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60F99: None of the above, but in this section

Coupon collector's problem higher asymptotics

This work is licensed under a Creative Commons Attribution 3.0 License.


Doumas, Aristides; Papanicolaou, Vassilis. Asymptotics of the rising moments for the coupon collector's problem. Electron. J. Probab. 18 (2013), paper no. 41, 15 pp. doi:10.1214/EJP.v18-1746. https://projecteuclid.org/euclid.ejp/1465064266

Export citation


  • Adler, Ilan; Oren, Shmuel; Ross, Sheldon M. The coupon-collector's problem revisited. J. Appl. Probab. 40 (2003), no. 2, 513–518.
  • Apostol, Tom M. Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976. xii+338 pp.
  • Baum, Leonard E.; Billingsley, Patrick. Asymptotic distributions for the coupon collector's problem. Ann. Math. Statist. 36 1965 1835–1839.
  • Bender, Carl M.; Orszag, Steven A. Advanced mathematical methods for scientists and engineers. I. Asymptotic methods and perturbation theory. Reprint of the 1978 original. Springer-Verlag, New York, 1999. xiv+593 pp. ISBN: 0-387-98931-5
  • Boneh, Arnon; Hofri, Micha. The coupon-collector problem revisited—a survey of engineering problems and computational methods. Comm. Statist. Stochastic Models 13 (1997), no. 1, 39–66.
  • Boneh, Shahar; Papanicolaou, Vassilis G. General asymptotic estimates for the coupon collector problem. J. Comput. Appl. Math. 67 (1996), no. 2, 277–289.
  • Brayton, Robert King. On the asymptotic behavior of the number of trials necessary to complete a set with random selection. J. Math. Anal. Appl. 7 1963 31–61.
  • Doumas, Aristides V.; Papanicolaou, Vassilis G. The coupon collector's problem revisited: asymptotics of the variance. Adv. in Appl. Probab. 44 (2012), no. 1, 166–195.
  • du Boisberranger, Jérémie; Gardy, Danièle; Ponty, Yann. The weighted words collector. 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), 243–264, Discrete Math. Theor. Comput. Sci. Proc., AQ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012.
  • Durrett, Richard. Probability: theory and examples. Second edition. Duxbury Press, Belmont, CA, 1996. xiii+503 pp. ISBN: 0-534-24318-5
  • Erdős, P.; Rényi, A. On a classical problem of probability theory. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 1961 215–220.
  • Feller, William. An introduction to probability theory and its applications. Vol. I. Third edition John Wiley & Sons, Inc., New York-London-Sydney 1968 xviii+509 pp.
  • Flajolet, Philippe; Gardy, Danièle; Thimonier, Lois. Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Appl. Math. 39 (1992), no. 3, 207–229.
  • Foata, Dominique; Han, Guo-Niu; Lass, Bodo. Les nombres hyperharmoniques et la fratrie du collectionneur de vignettes. (French) [Hyperharmonic numbers and the coupon collector's brotherhood] Sém. Lothar. Combin. 47 (2001/02), Article B47a, 20 pp.
  • Godwin, H. J.: On cartophily and motor cars, textitMath. Gazette 33 (1949) 169–171.
  • Holst, Lars. On birthday, collectors', occupancy and other classical urn problems. Internat. Statist. Rev. 54 (1986), no. 1, 15–27.
  • Janson, Svante. Limit theorems for some sequential occupancy problems. J. Appl. Probab. 20 (1983), no. 3, 545–553.
  • Neal, Peter. The generalised coupon collector problem. J. Appl. Probab. 45 (2008), no. 3, 621–629.
  • Newman, Donald J.; Shepp, Lawrence. The double dixie cup problem. Amer. Math. Monthly 67 1960 58–61.
  • Rudin, Walter. Real and complex analysis. Third edition. McGraw-Hill Book Co., New York, 1987. xiv+416 pp. ISBN: 0-07-054234-1